Abstract
Nonequilibrium dynamics governed by electron–phonon (eph) interactions plays a key role in electronic devices and spectroscopies and is central to understanding electronic excitations in materials. The realtime Boltzmann transport equation (rtBTE) with collision processes computed from first principles can describe the coupled dynamics of electrons and atomic vibrations (phonons). Yet, a bottleneck of these simulations is the calculation of e–ph scattering integrals on dense momentum grids at each time step. Here we show a datadriven approach based on dynamic mode decomposition (DMD) that can accelerate the time propagation of the rtBTE and identify dominant electronic processes. We apply this approach to two case studies, highfield charge transport and ultrafast excited electron relaxation. In both cases, simulating only a short time window of ~10% of the dynamics suffices to predict the dynamics from initial excitation to steady state using DMD extrapolation. Analysis of the momentumspace modes extracted from DMD sheds light on the microscopic mechanisms governing electron relaxation to a steady state or equilibrium. The combination of accuracy and efficiency makes our DMDbased method a valuable tool for investigating ultrafast dynamics in a wide range of materials.
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Introduction
Firstprinciples calculations are widely employed for modeling and designing materials, with applications ranging from energy^{1,2} to (opto)electronic devices^{3,4,5} to materials discovery^{6,7,8}. Starting from the crystal structure and atomic positions as the main inputs, these methods can predict material properties, including mechanical, electrical, magnetic, and optical. While computing groundstate and linearresponse properties with density functional theory (DFT) is a decadeslong effort^{9,10,11,12}, recent work has focused on modeling ultrafast dynamics in materials and simulating timedomain spectroscopies from first principles^{13,14,15,16,17,18,19,20,21,22,23}. These methods focused on nonequilibrium dynamics are a more recent research frontier with both theoretical and computational challenges.
Firstprinciples calculations in the time domain provide a microscopic description of nonequilibrium dynamics in materials. These methods propagate in time quantities characterizing the quantum dynamics, such as the timedependent electron wave function, density^{24}, density matrix^{25}, or Green’s function^{21}, and can also access the timedependent atomic positions and lattice vibrations^{17,18}. Different schemes are successful in different regimes. For example, coherent electron dynamics on the attosecond time scale can be modeled effectively using timedependent DFT^{22,26,27,28,29}, but that approach is not ideal for modeling phonon dynamics, which occurs on a picosecond time scale^{30}.
The realtime Boltzmann transport equation (rtBTE) has emerged as an effective tool for exploring the coupled electron and phonon dynamics from femtosecond to nanosecond timescales^{15,18,31}. In the rtBTE, the timedependent electron populations are obtained by solving a set of integrodifferential equations accounting for the e–ph scattering processes on dense momentum grids. Following an initial excitation, the rtBTE is propagated in time to reach thermal equilibrium or steady state in an external field. This scheme employs a femtosecond time step to capture the e–ph scattering processes. However, evaluating the scattering integral at each time step makes the rtBTE approach computationally demanding, even for materials with a handful of atoms in the unit cell.
Datadriven techniques are increasingly employed in materials modeling, both for accelerating computational workflows and to gain physical insight using learning algorithms^{32,33}. In particular, dynamic mode decomposition (DMD), which was developed in the last decade to study fluid dynamics, is a valuable tool to linearize dynamical problems and reduce their dimensionality^{34,35}. In DMD, explicit simulation of a short initial time window allows one to learn the dominant modes governing the dynamics and extrapolate the simulation to future times at low computational cost. Recent work has employed DMD to study electron dynamics described by model Hamiltonians with purely electronic interactions^{36,37}. Yet, to date, DMD has not been applied to more computationally intensive firstprinciples studies.
In this work, we combine DMD with firstprinciples calculations of nonequilibrium electron dynamics, using the framework of the rtBTE in the presence of e–ph collisions and external fields. We show that DMD provides an orderofmagnitude computational speedup while retaining the full accuracy of the firstprinciples rtBTE. In addition, DMD reveals key momentumspace temporal patterns and achieves a significant dimensionality reduction of the nonequilibrium physics. Our results include both highfield transport and transient excitedstate dynamics and are accompanied by a careful characterization of convergence with respect to the size of the sampling window during which DMD learns the dominant modes. Taken together, this work provides the blueprint for combining datadriven methods with firstprinciples calculations to study nonequilibrium dynamics in real materials.
Results
Firstprinciples rtBTE
We describe the electron distribution using the timedependent populations f_{nk}(t), which quantify the occupation of each electronic state \(\left\vert n{{{\bf{k}}}}\right\rangle\), where k is the electron crystal momentum, and n is the band index (from now on, we omit the band index to simplify the notation). Starting from an initial distribution at time zero, f_{k}(t = 0), in the rtBTE, the populations evolve according to^{38}
where \({{{\mathcal{I}}}}[{f}_{{{{\bf{k}}}}}(t)]\) is the collision integral accounting for eph scattering processes in momentum space^{39} and F includes any external fields applied to the system.
The rtBTE simulations use dense momentum grids to accurately describe scattering between electronic states via absorption and emission of phonons. The required grid sizes are typically >100 × 100 × 100 for both electron and phonon momenta. We timestep Eq. (1) using explicit solvers (Euler or fourthorder Runge–Kutta) or more advanced Strang splitting techniques^{31}. The collision integral includes a summation over the phonon momentum grid and is evaluated at least once per time step using a parallel algorithm implemented in the Perturbo code^{38} (see the “Methods” section for details). Although here we focus on the dynamics of electrons interacting with phonons, the rtBTE formalism has also been extended to study nonequilibrium phonon^{17} and exciton dynamics^{40}.
DMD learning and prediction of the dynamics
We employ DMD in combination with rtBTE simulations. The DMD approach linearizes the dynamics by relating the states of the system at times t and t + Δt via a timeindependent matrix A^{41,42}. Focusing on the e–ph dynamics, this amounts to advancing the electronic populations at time t using
where the populations f_{k} form a vector with size N equal to the number of kpoints in the electronic momentum grid (typically, N ≈ 10^{5}−10^{6}). To obtain the matrix A, we timestep the rtBTE in a sampling window consisting of M time steps (using the Perturbo code^{38}), and then we form two matrices X_{1} and X_{2}. The populations f_{k}(t) from t_{1} to t_{M−1} are stacked columnwise in the matrix X_{1}, with column i corresponding to time t_{i} and containing the populations f_{k}(t_{i}) for all kpoints and bands. The populations from t_{2} to t_{M} are similarly stacked columnwise in the second matrix X_{2}.
According to equation (2), these matrices are related by X_{2} = AX_{1}, but computing A naively from the pseudoinverse of X_{1} has a prohibitive cost due to the large size N of the kpoint grid. To circumvent this problem, in DMD, one first performs a truncated singular value decomposition (SVD)^{43,44} of the X_{1} matrix:
where \({{{\mathbf{\Sigma }}}}\in {{\mathbb{R}}}^{N\times (M1)}\) is a matrix with diagonal entries equal to the singular values σ_{j} arranged in decreasing order, while \({{{\bf{U}}}}\in {{\mathbb{C}}}^{N\times N}\) and \({{{\bf{V}}}}\in {{\mathbb{C}}}^{(M1)\times (M1)}\) are matrices collecting the mutually orthogonal singular vectors^{45}. (Above, V^{†} indicates the Hermitian conjugate of V).
Because X_{1} contains timedependent populations, this SVD procedure can single out the main patterns in the momentumspace dynamics. Here, we keep only the first r singular values (typically, r ≈ 10) to restrict the solution space to the leading r momentumspace modes and then project the matrix A onto this reduced rdimensional space. This procedure provides the matrix \(\tilde{{{{\bf{A}}}}}\), with reduced size r × r, which can be diagonalized straightforwardly to obtain the dominant DMD modes. Using this procedure, the populations at future times t > t_{M} are predicted—that is, obtained without explicit solution of the rtBTE—using
where \({\phi }_{{{{\bf{k}}}}}^{l}\) are the momentumspace DMD modes obtained from the matrix \(\tilde{{{{\bf{A}}}}}\), and \({\omega }_{l}^{{{{\rm{DMD}}}}}\) and b_{l} are their frequencies and amplitudes (see the “Methods” section for detailed derivations).
We summarize the main steps of this DMD procedure, which are illustrated in Fig. 1:

1.
Simulate the rtBTE dynamics for the first M steps and construct the matrices X_{1} and X_{2};

2.
Perform SVD on X_{1} to find the matrix \(\tilde{{{{\bf{A}}}}}\) in the reduced rdimensional space;

3.
Diagonalize \(\tilde{{{{\bf{A}}}}}\) to find the DMD modes \({\phi }_{{{{\bf{k}}}}}^{l}\) and their frequencies \({\omega }_{l}^{{{{\rm{DMD}}}}}\), with l = 1…r;

4.
Obtain the mode amplitudes b_{l} from the initial condition f_{k}(t_{1});

5.
Predict the dynamics for t > t_{M} using Eq. (4).
A key parameter is the duration of the sampling window (t_{M}) required for accurate DMD extrapolation of the dynamics beyond t_{M}. As the computational cost of DMD is negligible, the size of the sampling window, during which the rtBTE is solved by explicit timestepping, determines the computational cost of the entire workflow.
Highfield electron dynamics
We employ our DMDbased approach to simulate timedomain electron dynamics in an applied electric field in the presence of eph collisions. We recently demonstrated similar calculations using the rtBTE without the aid of datadriven techniques^{31}. Here, we use this case study to explore the accuracy and efficiency of our rtBTE plus DMD approach, as well as find optimal values for the sampling window and analyze the momentumspace DMD modes. Our calculations focus on electrons in GaAs, where the conduction band has three sets of lowenergy valleys, at Γ and near L and X in order of increasing energy^{46} (see the inset in Fig. 2a). Upon applying an electric field, the electrons are accelerated to higher band energies while they also transfer part of that excess energy to the lattice via e–ph collisions. These competing mechanisms lead to a steadystate electronic distribution which is typically reached on a picosecond to nanosecond time scale.
Our simulations begin with electrons in thermal equilibrium with the lattice at 300 K. We apply a constant electric field E and timestep the electron populations until they reach the steadystate distribution, \({f}_{{{{\bf{k}}}}}^{{{{\bf{E}}}}}\), from which we compute the mean drift velocity, v(E), a quantity routinely measured in experiments^{47,48,49}. Repeating this procedure for multiple field values allows us to construct the full drift velocity versus electric field curve in a material, starting from linear response at low field to velocity saturation at high field^{31}.
Figure 2a shows the timedependent populations in four regions of the Brillouin zone following the application of a high field (5 kV cm^{−1}). Electrons initially occupying the Γvalley scatter to the higherenergy L and Xvalleys. As a result, the electron populations in the Γvalley decrease, with a corresponding increase in L and Xvalley populations. In regions of momentum space between the Γ and Lvalleys, the populations peak at intermediate times and then relax to lower values.
This dynamics is nontrivial because the populations evolve differently in different momentumspace regions, making accurate predictions challenging. Our DMD approach can learn the dominant modes governing these intricate dynamics and extrapolate the timedependent populations well beyond the sampling window. Remarkably, we find that a short sampling window −400 fs to 2 ps out of a total simulation time of 12.5 ps —is sufficient to extrapolate the dynamics all the way to steady state, with rtBTE and DMD trajectories in nearly exact agreement outside the sampling window (Fig. 2a). This accuracy extends to the entire set of ~10^{5} kpoints considered in our simulations, providing carrier number conservation within 1% error.
The ability to learn key temporal momentumspace patterns is a consequence of the relatively rapid decay of the singular values of the X_{1} matrix used for learning the dynamics in the sampling window (Fig. 2b). This decay becomes slower as the sampling window increases, but it remains significant even for the longest sampling window of 2 ps used here (note the log scale in the plot). In turn, the singular value decay enables a striking dimensionality reduction, with DMD employing only r ≈ 10 modes to solve the dynamics as opposed to 10^{5} populations f_{k} and billions of e–ph scattering terms in the rtBTE.
The choice of an ideal sampling window can rely on the appearance of specific DMD modes at a steady state. Figure 2c shows the DMD mode frequencies ω^{DMD} in the complex plane, where the imaginary part of ω^{DMD} corresponds to the decay rate of a given mode, and the real part gives its oscillation frequency. The populations f_{k}(t) are realvalued and are written as a summation of complex exponentials in Eq. (4). Therefore, physically meaningful results are possible only when \({{{\rm{Re}}}}({\omega }^{{{{\rm{DMD}}}}})=0\) (modes 1, 2) or when ω^{DMD} appear as conjugate pairs (modes 3−10). Describing the steady state is particularly important in our simulations. In DMD, all modes with a nonzero imaginary frequency vanish in the long time limit, with only one mode surviving at steady state (mode 1 in Fig. 2c). As the sampling window increases, the imaginary frequency of this mode goes to zero, providing the correct steadystate behavior. This analysis allows us to find the minimal sampling window required for accurate steadystate results by monitoring the zerofrequency mode.
The DMD eigenvector of the zerofrequency mode (mode 1 in Fig. 2d) determines the steadystate electron distribution \({b}_{l}{\phi }_{{{{\bf{k}}}}}^{1}={f}_{{{{\bf{k}}}}}(t\to \infty )\), while the other modes control the transient dynamics. For example, mode 2 governs electron scattering from the Γ to the L and Xvalleys, and higher modes appearing as conjugate pairs exhibit oscillating trends in energy (modes 3−10 in Fig. 2d). Converging the zerofrequency mode allows us to compute the steadystate drift velocity more efficiently. Figure 2e shows that a sampling window of 1.7 ps (170 snapshots) provides a drift velocity nearly identical to the full rtBTE calculation, which requires much longer simulation times of up to 12.5 ps (1250 snapshots). On this basis, we conclude that DMD needs only ~10% of the dynamics data for accurate steadystate predictions.
Velocityfield curves
We also employ DMD to accelerate calculations of entire velocityfield curves. This requires the drift velocity for a set of electric field values, and thus we adopt a modified workflow. Following our recent work^{31}, we gradually increase the electric field (black curve in Fig. 3a) and use the steadystate populations for a given field, \({f}_{{{{\bf{k}}}}}^{{{{\bf{E}}}}}\), as the initial condition for the next field value, E + ΔE, where the field increment ΔE is typically 100−200 V cm^{−1}. As the applied field increases, the DMD frequencies and momentumspace modes change substantially. Therefore, for each new field value, we repeat DMD learning in the initial stage of the simulation (see the DMD sampling regions shown as red rectangles in Fig. 3a). We then predict the steadystate populations using mode 1 from DMD, \({f}_{{{{\bf{k}}}}}^{{{{\bf{E}}}}}={b}_{1}{\phi }_{{{{\bf{k}}}}}^{1}\) and the drift velocity for that field value, and use \({f}_{{{{\bf{k}}}}}^{{{{\bf{E}}}}}\) as the initial condition for the next field value.
The velocityfield curves obtained with this approach are shown in Fig. 3b for GaAs and graphene and compared with rtBTE results obtained without DMD. Using DMD lowers significantly the computational cost to obtain the full velocityfield curves by a factor of 10.5 for GaAs and ~ 16.5 for graphene, while fully preserving the accuracy. Because the drift velocity is computed as a weighted sum of \({f}_{{{{\bf{k}}}}}^{{{{\bf{E}}}}}\)^{31}, the nearly exact agreement between the DMD and full rtBTE results demonstrates the accuracy of the DMD populations in momentum space. The DMD efficiency is a consequence of its ability to capture the dominant modes in the population dynamics using only a small number of snapshots, with a similar accuracy regardless of the electric field value. Our strategy of gradually increasing the electric field leads to easiertoextrapolate dynamics compared to the abrupt application of a strong field.
Excited electron relaxation
Next, we consider different nonequilibrium dynamics where the material is initially prepared in an excited electronic state. This setting can be used, for example, to model the effect of an optical excitation with a laser pulse^{13}. Different from the highfield dynamics, in this case, the longtime limit is known, and we are primarily interested in the transient dynamics. Following the initial excitation, in the presence of e–ph interactions and without any external fields, the electrons relax to a thermal equilibrium Fermi–Dirac distribution^{50}, \({f}_{{{{\bf{k}}}}}^{{{{\rm{FD}}}}}\), typically on a subpicosecond time scale. This ultrafast dynamics can be modeled by timestepping the rtBTE until reaching the equilibrium FermiDirac distribution. Using this approach, our previous work has shown that electrons relax to the band edge significantly slower than holes in GaN semiconductors, with implications for optoelectronic devices^{15}.
Following that work, we model an excited state in GaN by placing the electrons ~1 eV above the conduction band edge and then obtain the timedependent electron populations by solving the rtBTE (see Fig. 4a, b). We employ DMD to predict this transient dynamics and find large errors when using a short time window of up to ~50 fs (solid red line in Fig. 4c). The correct steady state and transient dynamics are obtained by increasing the sampling window to 200 fs (dashed orange line in Fig. 4c). Our analysis of the DMD frequencies shows that the zerofrequency mode describing thermal equilibrium in the longtime limit appears when the sampling window reaches 100 fs (see the arrow in Fig. 4d) and fully converges for a ~200 fs sampling window. The need for such a long sampling window relative to the total duration of the dynamics (400 fs) makes DMD ineffective.
To address this issue and more efficiently study transient dynamics with DMD, we formulate a different learning procedure that incorporates knowledge of the equilibrium state. We focus on the difference between the transient and equilibrium populations, \(\delta {f}_{{{{\bf{k}}}}}(t)={f}_{{{{\bf{k}}}}}(t){f}_{{{{\bf{k}}}}}^{{{{\rm{FD}}}}}\), as opposed to just f_{k}(t) as we did in the highfield example. After predicting δf_{k}(t) with DMD, we obtain the timedependent populations f_{k}(t) by adding back the \({f}_{{{{\bf{k}}}}}^{{{{\rm{FD}}}}}\) term. As δf_{k} vanishes in the longtime limit (Fig. 4b), the zerofrequency DMD mode is missing when computing δf_{k} (Fig. 4e); all other DMD frequencies associated with δf_{k} are similar to those for f_{k}(t) (Fig. 4d, e). We find that the DMD method based on δf_{k} is far more effective and requires a significantly shorter sampling window for accurate DMD predictions−using a 50 fs sampling window, we achieve results similar to DMD for f_{k}(t) with a four times longer (200 fs) window (Fig. 4c).
With this improved DMD approach, using a sampling window of only ~12% of the total simulation time allows us to accurately predict the average electron relaxation rate in GaN, with a DMD computed value of 5.23 eV fs^{−1} in close agreement (within 0.8%) with the rtBTE result. This result demonstrates that our DMD approach can predict excited electron relaxation with a high accuracy.
Discussion
The DMD approach introduced here is very efficient: the rtBTE is solved explicitly on a highperformance computer only for a small number of initial time steps, after which the entire dynamics can be computed straightforwardly with DMD, using only a laptop. The most demanding step is carrying out truncated SVD on the X_{1} matrix, but for comparison, this step requires lower computational resources than even just a single rtBTE time step.
This remarkable speedup is achieved by reducing the dimensionality of the rtBTE dynamics and is linked to the shape of the X_{1} matrix. The rtBTE employs a large number of kpoints (about 10^{5}−10^{6}), which equals the number of rows of the matrix X_{1}, and a significantly smaller number of snapshots in the DMD sampling window, typically ~100 time steps, which sets the number of columns in X_{1}. Following truncated SVD, the size of the problem is reduced to (at most) the number of snapshots and is typically of order 50−100, and thus smaller by orders of magnitude compared to the original rtBTE. (Note that one could use the entire set of singular values, but here we prefer using only ~10 singular values to prevent numerical instabilities^{45}).
This efficiency allows us to evaluate the accuracy of DMD on the fly, halting explicit timestepping of the rtBTE when the DMD steady state or transient dynamics are fully converged. In addition, our approach addresses the key challenge of storing the rtBTE populations. This is a critical improvement because in conventional rtBTE simulations one needs to store the populations f_{k}(t) on dense momentum grids for thousands of time steps, resulting in terabytes of data. In contrast, after carrying out SVD in the sampling window, DMD stores only a handful of complex frequencies and momentumspace modes, using which the dynamics can be reconstructed for the entire simulation.
In summary, we have introduced a datadriven approach based on DMD to accelerate firstprinciples calculations of nonequilibrium electron dynamics in materials. Our method speeds up the solution of the timedependent Boltzmann equation with electron collisions computed from first principles. We have shown that DMD can capture dominant modes governing the microscopic dynamics, enabling accurate predictions of the steadystate properties such as the drift velocity as well as transient processes such as electron relaxation and equilibration. In both steadystate and transient nonequilibrium calculations, DMD requires explicit timestepping of the rtBTE in a time window of only ~10% of the full simulation, after which the dynamics is extrapolated from the DMD modes with negligible computational cost. This DMD workflow preserves the accuracy while requiring far more modest computational resources than full rtBTE simulations.
These advances are broadly relevant to studying nonequilibrium quantum dynamics of elementary excitations. For example, in future work, our datadriven approach will be extended to study the coupled dynamics of electrons and phonons, which involves fast (electron) and slow (phonon) timescales. The current DMD approach is not designed to address such multiscale nonequilibrium dynamics, and extensions using multiresolution DMD will be explored.
Methods
Computing DMD modes and frequencies
Let us describe in more detail the calculation of DMD modes and frequencies. We start from the snapshots f_{k}(t) evaluated explicitly with the rtBTE in the sampling window t_{1} < t < t_{M}, and then apply the SVD procedure to the matrix X_{1} (see Eq. (3)). As shown in Fig. 2b, we find that the singular values σ_{j} decay rapidly. Keeping only the largest r ≈ 10 singular values, we write the SVD of X_{1} as
where we defined the economysized matrices in the rdimensional subspace^{45} as \(\tilde{{{{\mathbf{\Sigma }}}}}={{{\mathbf{\Sigma }}}}(1:r,1:r)\), \(\tilde{{{{\bf{U}}}}}={{{\bf{U}}}}(1:N,1:r)\), \(\tilde{{{{\bf{V}}}}}={{{\bf{V}}}}(1:M1,1:r)\). This way, the approximate pseudoinverse of the matrix X_{1}, denoted as \({{{{{\bf{X}}}}}_{1}}^{+}\), can be obtained with little effort as \(\tilde{{{{\bf{V}}}}}{\tilde{{{{\mathbf{\Sigma }}}}}}^{1}{\tilde{{{{\bf{U}}}}}}^{{\dagger} }\). Then the matrix A relating the snapshot matrices via X_{2} = AX_{1} can be written as
Note that the matrix A depends on the sampling window. Due to its large N × N size (here, N ≈ 10^{5} is the number of kpoints), diagonalizing A is computationally expensive. In DMD, a key step is rewriting this matrix in the reduced rdimensional space:
allowing for straightforward eigenvalue decomposition:
where the matrix W contains the eigenvectors of \(\tilde{{{{\bf{A}}}}}\) and the eigenvalues Λ = diag{λ_{l}} are common to both matrices \(\tilde{{{{\bf{A}}}}}\) and A^{51}. The DMD modes, stacked columnwise in the matrix \({{{\mathbf{\Phi }}}}=\left({{{{\boldsymbol{\phi }}}}}^{1}\ {{{{\boldsymbol{\phi }}}}}^{2}\cdots {{{{\boldsymbol{\phi }}}}}^{r}\right)\in {{\mathbb{C}}}^{N\times r}\), can be obtained using^{51}
The DMD frequency of mode l is obtained from the corresponding eigenvalue λ_{l} using Eq. (8),
where Δt is the simulation time step. To circumvent the potential addition of a \(2\pi m\,{{{\rm{i}}}},m\in {\mathbb{Z}}\) term due to \(\ln {\lambda }_{l}\) computation, we evaluate the logarithm in the following way: \(\ln {\lambda }_{l}=\ln  {\lambda }_{l} +{{{\rm{i}}}}\arg ({\lambda }_{l})\), with \(\arg ({\lambda }_{l})\) in (−π, π].
The mode amplitudes \({{{\bf{b}}}}=\left({b}_{1}\ {b}_{2}\cdots {b}_{r}\right)\in {{\mathbb{C}}}^{r}\) are obtained from the initial condition. Setting t = 0 in Eq. (4), we get
and thus the mode amplitude vector b is obtained from the pseudoinverse of the DMD mode matrix Φ:
The pseudoinverse of the matrix Φ is computed using truncated SVD and has a negligible computational cost compared to SVD of the X_{1} matrix due to (N, r) dimensions of the matrix Φ.
This approach provides the DMD modes \({\phi }_{{{{\bf{k}}}}}^{l}\), frequencies \({\omega }_{l}^{{{{\rm{DMD}}}}}\), and mode amplitudes b_{l}, and thus all the quantities needed for DMD prediction of the dynamics outside the sampling window (t > t_{M}) using Eq. (4).
Electron–phonon scattering from first principles
Our firstprinciples calculations of e–ph scattering employ an established workflow, which is summarized here and described in more detail in ref. ^{38}. The electronic wave functions and band energies are obtained from planewave DFT calculations with the Quantum Espresso code^{52} using the local density approximation^{53} and normconserving pseudopotentials^{54}. The electronic quasiparticle band structure is refined using GW calculations carried out with the, Yambo code^{55}. This step improves the agreement with the experiment of the electron effective masses and relative valley energies, which are essential for precise calculations of highfield dynamics^{31} and excited electron relaxation^{15}.
The phonon dispersion and eph perturbation potentials are obtained from densityfunctional perturbation theory (DFPT), where lattice vibrations and their coupling with electrons are treated as perturbations to the groundstate electron density^{52}. The e–ph interactions are described by the matrix elements
which are the probability amplitudes to scatter from an initial electronic state \(\left\vert {\psi }_{n{{{\bf{k}}}}}\right\rangle\), with band n and momentum k, to a final state \(\vert {\psi }_{m{{{\bf{k}}}}+{{{\bf{q}}}}}\rangle\) by absorbing or emitting a phonon with mode ν, momentum q, and frequency ω_{νq}. The term Δ_{νq}V^{KS} is the e–ph perturbation potential induced by the phonon mode and is defined in ref. ^{38}.
To study nonequilibrium dynamics with the rtBTE, the electrons, phonons, and e–ph scattering are described on dense k and qpoint momentum grids. Obtaining the e–ph matrix elements on such grids directly from DFPT is computationally prohibitive. Therefore, we first compute g_{mnν}(k, q) on coarse k and qpoint grids^{15,56} and then interpolate these quantities to significantly finer grids using Wannier–Fourier interpolation with Wannier functions generated from Wannier90^{57}. Finally, the e–ph scattering integral employed in Eq. (1) is defined as
where \({{{{\mathcal{N}}}}}_{{{{\bf{q}}}}}\) is the number of qpoints, ε_{nk} and ε_{mk+q} are the band energies of the initial and final electronic states, and the Dirac delta functions expressing energy conservation are implemented as Gaussians with a small (~5 meV) broadening. Above, F_{abs} and F_{em} are phonon absorption and emission terms, whose explicit expressions are given in ref. ^{38}. The Wannier interpolation, scattering integral computation, and the rtBTE ultrafast dynamics are implemented in our Perturbo opensource package^{38}.
Data availability
All the data supporting the results of this study are available upon reasonable request.
Code availability
The Perturbo code used in this work is opensource software and can be downloaded at https://perturbocode.github.io.
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Acknowledgements
This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, and Office of Basic Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC) program under Award No. DESC0022088. The ultrafast carrier dynamics calculations are based on work performed within the Liquid Sunlight Alliance, which is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, and Fuels from Sunlight Hub under Award DESC0021266. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DEAC0205CH11231.
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I.M., J. Yin, and J. Yao conducted the research work under the guidance of C.Y. and M.B. All authors contributed to writing the manuscript.
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Maliyov, I., Yin, J., Yao, J. et al. Dynamic mode decomposition of nonequilibrium electronphonon dynamics: accelerating the firstprinciples realtime Boltzmann equation. npj Comput Mater 10, 123 (2024). https://doi.org/10.1038/s41524024013084
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DOI: https://doi.org/10.1038/s41524024013084